Semilattice orders on the homomorphic images of the Rédei semigroup

Abstract

The combinatorial simple principal ideal semigroups generated by two elements were described by L. Megyesi and G. Pollak. The ‘most general’ among them is called the Redei semigroup. The ‘most special’ combinatorial simple principal ideal semigroup generated by two elements is the bicyclic semigroup. D. B. McAlister determined the compatible semilattice orders on the bicyclic semigroup. Our aim is to study the compatible semilattice orders on the homomorphic images of the Redei semigroup. We prove that there are four compatible total orders on these semigroups. We show that on the Redei semigroup, the total orders are the only compatible semilattice orders. Moreover, on each proper homomorphic image of the Redei semigroup, we give a compatible semilattice order which is not a total order.